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Transcript
Introduction to Planning
Han Yu
University of Central Florida
Outline

What is planning?

Formal definitions of planning problems

Basic planning algorithms

Recent development
What is Planning?

Planning is a search problem that requires to
find an efficient sequence of actions that
transform a system from a given starting
state to the goal state
What’s Given?

Initial state of the problem

Goal state of the problem

A finite set of actions:



pre-conditions: a finite set of conditions for
the action to be performed
post-conditions: a finite set of conditions
that will be changed after the action is
performed
cost
What’s Output?

A sequence of actions that meet the
following criteria



every action matches the current system
state
can transform system from initial state to
goal state
the total cost of the actions is below a
specified value
Planning as a Real-World Problem

Planning problem has a wide range of
applications in the real world

planning in daily life

game world

workflow management
Planning a Trip
Begin
Preparation
Rental Car Reservation
Confirm Reservations
Airline Reservation
Hotel Reservation
End
Towers of Hanoi
Problem Analysis



Optimal solution contains 2n-1 actions,
where n is the number of disks
Up to 6 possible actions in each system
state
The upper bound on search space to
n-1)
(2
find an optimal solution will be 6
Sliding-Tile Puzzle
Planning in Workflow Management
Start Assembly
Examine Order
Gather
Components
Assembly Box
and Motherboard
Install
Video Card
Install
Network Card
Insert Modem
Plug in CD
Plug in Battery
Install
Internal Disk
Install
Motherboard
Test
End Assembly
Partial-Order Plan versus
Total-Order Plan

Partial-order plan

consists partially ordered set of actions

sequence constraints exist on these actions


plan generation algorithm can be applied to
transform partial-order plan to total-order plan
Total-order plan

consists totally ordered set of actions
Partial-Order Plan
Get brush
Paint ceiling
Start
Get ladder
Finish
Total-Order Plan
Start
Get ladder
Get brush
Paint ceiling
Finish
Start
Get brush
Get ladder
Paint ceiling
Finish
Features of Planning Problems

Large search space

Action is associated with system states

Restrictions on the action sequence

Valid solution may not exist

Optimization requirement
STRIPS Planning System

A tuple T = (P, O, I, G), where




P is a finite set of ground literals, the conditions
O is a finite set of operators
I is the initial state, a subset of P
G is the goal state, a subset of P
STRIPS Operators

Each operator O has the following attributes

PC, a set of ground literals, defines the precondition of
the operator

D, a set of ground literals, defines the conditions that
will be removed after the operation is executed

A, a set of ground literals, defines the conditions that
will be added after the operation is executed

C, the cost of the operation
A Simple Example - Blocks World
A
C
A
B
B
Initial State
On(C, A)
Clear(Fl)
On(A, Fl)
Clear(B)
On(B, Fl)
Clear(C)
C
Goal State
On(A, B)
On(B, C)
On(C, Fl)
Clear(A)
Clear(Fl)
Graphical Representation of
Initial State
On(C, A)
Clear(Fl)
On(A, Fl) Clear(B)
Start
T
On(B, Fl)
Clear(C)
Graphical Representation of
Goal State
Nil
finish
On(A, B)
On(B, C)
On(C, Fl)
Clear(A)
Clear(Fl)
Block World - Operator

Move(x, y, z)

Move block x that is above y to above z

PC: On(x,y), Clear(x), Clear(z)

D: Clear(z), On(x, y)

A: On(x,z), Clear(y), Clear(Fl)
Graphical Representation of
Operator
A
Clear(y)
Operator
PC
On(x, y)
On(x, z)
Clear(Fl)
Move(x, y, z)
Clear(x)
Clear(z)
Forward Chaining


Search from the initial state
Expand the search tree by finding the set of all
applicable operators from the current state

applicable means the precondition of the operator is a
subset of current state



Update current state
For every state that is reached, record the shortest
path (or path with lowest cost) from the initial state
to this state
If the goal state is reached, stop the algorithm
Forward Chaining
On(C, Fl)
Clear(Fl)
On(A, Fl)
Clear(B)
On(B, Fl)
Clear(C)
Clear(A)
Move(C, A, Fl)
On(C, A)
Clear(Fl)
On(A, Fl)
Clear(B)
On(B, Fl)
On(B, C)
Clear(C)
Clear(Fl)
Move(B, Fl, C)
On(C, A)
Clear(B)
On(A, Fl)
Backward Chaining


Search backward from the goal state
Expand the search tree by finding the set of all
applicable operators that can reach the current
state

applicable means set A of the operator is a subset of
current state




Update the state
If the initial state is reached, stop the algorithm
The solution is a partial-ordered plan
Constraints in action ordering may be violated
Backward Chaining
On(A, B)
On(B, C)
On(C, Fl)
Clear(A)
Clear(Fl)
On(B, C)
On(C, Fl)
Clear(A)
Clear(Fl)
On(A, Fl)
Clear(B)
Move(A, Fl, B)
Recent Developments

Plan Reuse

Graphplan

Problem-specific planning

Evolutionary computation approach
Plan Reuse


Reuse old plans for new planning problems
Consists of two steps



plan matching
plan modification
Research findings



generally, plan reuse is even harder than plan from
scratch
do better only when two problems are close enough
plan matching could be the bottleneck
Graphplan


Partial-order general planner
Constructing a planning graph before search



plan graph contains all possible actions that can be
taken in each time step
actions that interfere with one another can coexist in
the graph
More efficient than other general planners in
some problems
Problem-specific Planning

Heuristics combined during search




heuristics is problem dependent
cannot apply to other problems
Usually outperforms general planners in
specific problems
Example

in sliding-tile puzzle, accurate estimation of the
distance between current state and goal state can
speed up the search for a plan
Genetic Algorithms



Parallel search and optimization
algorithm
Inspired by the basic rule of natural
selection, survival-of-the-fittest
Non-deterministic algorithm

randomness is incorporated during
implementation
Solution Formation

Candidate solution is encoded as a list of
genes, called chromosome
1


0
1
1
0
1
Start with a population of randomly
generated chromosomes
Population is evolved in every generation



evaluate each chromosome
select chromosomes to the next generation
apply genetic operations
Genetic Operation

crossover
Before
crossover
After
crossover
Crossover point
1
0
1
1
0
1
0
0
0
1
1
1
1 0 1 1 1 1
0
0
0
1
0
1
Genetic Operation

Mutation
Before
mutation
1
0
1
1
0
1
After
mutation
1
0
0
1
0
1
GA Procedure

Initialize the population

Evaluate each chromosome in the population

While the stopping condition is not met


select chromosomes to the next generation

crossover, mutation

evaluate newly generated chromosomes
End while
Evolutionary Computation
Approach




Non-deterministic algorithm
Starts from a set of randomized plan
Plans are evolved during generations
In each generation



evaluate the performance of the plan
select the plans to next generation, based on
performance
crossover, mutation to create new plans
Evolutionary Computation
Approach

Solution encoding


a list of floating-point number, each number maps
to an action in the problem
number of actions can vary during search
0.8 0.5 0.3 0.2 0.5
A4 A3 A2 A1 A2
Evolutionary Computation
Approach

State-aware crossover

crossover points are selected based on the
matching of states
s1
Parent 1
s2
Parent 2
Child 1
Child 2
Match(s1, s2) = true
Evolutionary Computation
Approach

Mutation

randomly select an action and replace it with a
random floating-point number
Before
mutation
0.8 0.5 0.3 0.2 0.5
After
mutation
0.8 0.5 0.6 0.2 0.5
Evolutionary Computation
Approach

Fitness evaluation


consists of two parts: goal fitness fg and cost
fitness fc
goal fitness: how the solutions reach the goal
state

cost fitness: the total cost of the solution

overall fitness = a * fg + b * fc
Experimental Results

Towers of Hanoi



Sliding-tile puzzle




can solve the 5 and 6-disk case with high
probability
state-aware crossover does not perform well
can solve 3*3 problem case with high probability
state-aware crossover perform much better since
a looser definition of matching state is used
Performance scalability is not good
Heuristics may be helpful to improve the
References

Artificial Intelligence: A Modern Approach. Stuart
Russell and Peter Norvig. Prentice Hall.

Artificial Intelligence: A New Symthesis. Nils Nilsson.
Kaufmann.


Intelligent Planning: A Decomposition and Abstraction
Based Approach. Qiang Yang. Springer.
Internet-Based Workflow Management: Towards a
Semantic Web. Dan C. Marinescu. Wiley, 2002.
References

Plan reuse versus plan generation: a theoretical and
empirical analysis. B. Nebel, J. Koehler. Journal of Artificial
Intelligence 76 (1995), 427-454.

Fast planning through planning graph analysis. A. L. Blum,

M. L. Furst Journal of Artificial Intelligence 90 (1997), 281300.
Planning as heuristic search. B. Bonet, H. Geffner. Journal of
Artificial Intelligence 129 (2001), 5-33.
What it is, What it could be, An introduction to the Special Issue

on Planning and Scheduling. D. McDermott, J. Handler.
Journal of Artificial Intelligence 76 (1995), 1-16.
Concluding Words
Failing to plan is planning to fail
--Effie Jones